How to Factor Polynomials?

Knowing how to factor polynomials is a basis of Algebra 1 and beyond. Factoring undoes multiplication and lets you solve equations, simplify expressions, and analyze graphs. This page covers the most important factoring strategies, greatest common factor (GCF), difference of squares, perfect square trinomials, and factoring by grouping, with clear examples and two comprehensive video lessons.

What Does It Mean to Factor a Polynomial?

Factoring a polynomial means rewriting it as a product of simpler expressions (its factors) whose product equals the original polynomial. For example, \(\color{blue}{6x^{3} + 9x^{2} = 3x^{2}(2x + 3)}\). You can verify any factoring answer by multiplying the factors back together.

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Strategy 1: Factor Out the Greatest Common Factor (GCF)

Always look for a GCF first. The GCF is the largest monomial that divides evenly into every term.

• \(\color{blue}{6x^{3} + 9x^{2}}\): GCF = \(\color{blue}{3x^{2}}\) → \(\color{blue}{3x^{2}(2x + 3)}\)
• \(\color{blue}{4x^{2} – 12x}\): GCF = \(\color{blue}{4x}\) → \(\color{blue}{4x(x – 3)}\)

Strategy 2: Difference of Two Squares

If a binomial has the form \(\color{blue}{a^{2} – b^{2}}\), it factors as \(\color{blue}{(a + b)(a – b)}\).

• \(\color{blue}{x^{2} – 16 = (x + 4)(x – 4)}\)
• \(\color{blue}{9x^{2} – 25 = (3x + 5)(3x – 5)}\)

Strategy 3: Perfect Square Trinomial

Recognize \(\color{blue}{a^{2} + 2\text{ ab } + b^{2} = (a + b)^{2}}\) and \(\color{blue}{a^{2} – 2\text{ ab } + b^{2} = (a – b)^{2}}\).

• \(\color{blue}{x^{2} + 6x + 9 = (x + 3)^{2}}\)

Strategy 4: Factoring Trinomials (ax² + \(\color{blue}{\text{ bx } + c}\))

For \(\color{blue}{a = 1}\), find two numbers with product c and sum b. For a > 1, use the AC method (see the Factoring Trinomials page).

Strategy 5: Factoring by Grouping

For four-term polynomials, group the terms in pairs, factor each pair, then factor out the common binomial.

• \(\color{blue}{x^{3} + 2x^{2} + 3x + 6}\) → \(\color{blue}{x^{2}(x + 2) + 3(x + 2)}\) → \(\color{blue}{(x^{2} + 3)(x + 2)}\)

Step-by-Step Summary

1. Factor out the GCF first (always).
2. Count the remaining terms: 2 terms → difference of squares or sum/difference of cubes; 3 terms → trinomial; 4 terms → grouping.
3. Apply the appropriate strategy.
4. Check each factor to see if it can be factored further.
5. Verify by multiplying back.

Watch: How to Factor Polynomials (Video Lesson)

The Organic Chemistry Tutor covers the most common factoring strategies clearly and efficiently:

Factoring Polynomials — Worked Examples

Example 1: Factor \(\color{blue}{6x^{3} + 9x^{2}}\).

GCF = \(\color{blue}{3x^{2}}\). Divide each term: \(\color{blue}{6x^{3} \div 3x^{2} = 2x}\); \(\color{blue}{9x^{2} \div 3x^{2} = 3}\).
Answer: \(\color{blue}{3x^{2}(2x + 3)}\)

Example 2: Factor \(\color{blue}{x^{2} – 16}\).

Difference of squares: \(\color{blue}{a = x}\), \(\color{blue}{b = 4}\).
Answer: \(\color{blue}{(x + 4)(x – 4)}\)

Example 3: Factor \(\color{blue}{x^{2} + 6x + 9}\).

Perfect square: \(\color{blue}{\sqrt{9} = 3}\); middle term \(\color{blue}{2(x)(3) = 6x}\) matches.
Answer: \(\color{blue}{(x + 3)^{2}}\)

Example 4: Factor \(\color{blue}{x^{3} – x^{2} – 6x}\).

\(\color{blue}{\text{ GCF } = x}\): \(\color{blue}{x(x^{2} – x – 6)}\). Factor the trinomial: product = −6, sum = −1 → −3 and 2.
Answer: \(\color{blue}{x(x – 3)(x + 2)}\)

More Factoring Strategies: Comprehensive Video

This in-depth video covers GCF, AC method, grouping, and more, great for a full review:

Exercises for Factoring Polynomials

Factor each polynomial completely.

1. \(\color{blue}{8x^{2} + 12x}\)
2. \(\color{blue}{x^{2} – 25}\)
3. \(\color{blue}{x^{2} + 6x + 9}\)
4. \(\color{blue}{2x^{3} – 8x}\)
5. \(\color{blue}{x^{3} – x^{2} – 6x}\)
6. \(\color{blue}{3x^{2} – 12}\)

Answers

1. \(\color{blue}{4x(2x + 3)}\)
2. \(\color{blue}{(x + 5)(x – 5)}\)
3. \(\color{blue}{(x + 3)^{2}}\)
4. \(\color{blue}{2x(x + 2)(x – 2)}\)
5. \(\color{blue}{x(x – 3)(x + 2)}\)
6. \(\color{blue}{3(x + 2)(x – 2)}\)

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Free Factoring Polynomials Worksheet

Ready to practice on your own? Download our free Factoring Polynomials worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Factoring Polynomials before a quiz or test.

Download Greatest Common Factor and GCF Factoring Worksheet

Frequently Asked Questions

What is the first step in factoring any polynomial?

Always factor out the Greatest Common Factor (GCF) first. This simplifies what remains and makes the subsequent factoring steps much easier.

How do I know which factoring method to use?

Count the terms: 2 terms suggest difference of squares; 3 terms suggest trinomial factoring; 4 terms suggest grouping. After removing the GCF, match the pattern of the remaining expression.

What if a polynomial does not factor?

Some polynomials are prime. They cannot be factored over the integers. You can confirm by checking the discriminant for quadratics or by testing all factor pairs systematically.

Related Topics

• Factoring Trinomials
• Perfect Square Trinomial
• Multiplying Binomials
• Simplifying Polynomials
• Multiplying and Dividing Monomials